
where \(N_\text{sub}\)  is the number of overlapping subdomains \(\Omega _s\)  into which the domain \(\Omega \)  is decomposed. To define the matrices present in (REF ), let \(\mathcal {N}\)  be an ordered set of the unknowns of the whole domain and let \(\mathcal {N} = \bigcup _{s=1}^{N_\text{sub}} \mathcal {N}_s\)  be its decomposition into the non-disjoint ordered subsets corresponding to the different overlapping subdomains \(\Omega _s\) . Further, define \(N = \vert \mathcal {N} \vert \)  and \(N_s = \vert \mathcal {N}_s \vert \) . The \(N_s \times N_s\)  matrix \(A_s\)  stem from the discretisation of local boundary value problems on \(\Omega _s\)  with transmission conditions at the subdomain interfaces. The \(N_s \times N\)  matrix \(R_s\)  is the Boolean restriction matrix from \(\Omega \)  to subdomain \(\Omega _s\)  while \(R_s^T\)  is then the extension matrix from subdomain \(\Omega _s\)  to \(\Omega \) . The \(N_s \times N_s\)  diagonal matrices \(D_s\)  provide a discrete partition of unity, i.e., are such that \(\sum _{s=1}^{N_\text{sub}} R_s^T D_s R_s = I\) . See, e.g., [1]}, [2]} for further details on such methods. PMLs are introduced as transmission conditions on the interface boundaries of the local subdomains in [3]}. In this approach, the PML region is included strictly inside the overlap, the PML region being the outermost layers within each overlapping domain. This ensures that there is enough overlap for the approach to be efficient and sufficient length of the PML for a good approximation of the interface transmission condition.
